How did Classic Greeks understand natural numbers?

A commentary to Plato’s Charmaides by unknown scholar describes a clear distinction between ideal natural numbers and numbers used for mundane everyday counting:

Logistic is the science that treats of numbered objects, not of numbers; it does not consider number in the true sense, but it works with 1 as unit and the numbered object as number, e.g. it regards 3 as a triad and 10 as a decad, and applies the theorems of arithmetic to such cases. It is, then, logistic which treats on the one hand the problem called by Archimedes the cattle-problem, and on the other hand melite and phialite numbers, the latter appertaining to bowls, the former to flocks; in other types of problem too it has regard to the number of sensible bodies, treating them as absolute. Its subject-matter is everything that is numbered; its branches include the so-called Greek and Egyptian methods in multiplications and divisions, as well as the addition and splitting up of fractions, whereby it explores the secrets lurking in the subject-matter of the problems by means of the theory of triangular and polygonal numbers. Its aim is to provide a common ground in the relations of life and to be useful in making contracts, but it appears to regard sensible objects as though they were absolute.